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Ecology, 91(11), 2010, pp. 3384–3397
Ó 2010 by the Ecological Society of America
Null model analysis of species associations using abundance data
WERNER ULRICH1,3
1
AND
NICHOLAS J. GOTELLI2
Department of Animal Ecology, Nicolaus Copernicus University, Gagarina 9, 87-100 Toruń, Poland
2
Department of Biology, University of Vermont, Burlington, Vermont 05405 USA
Abstract. The influence of negative species interactions has dominated much of the
literature on community assembly rules. Patterns of negative covariation among species are
typically documented through null model analyses of binary presence/absence matrices in
which rows designate species, columns designate sites, and the matrix entries indicate the
presence (1) or absence (0) of a particular species in a particular site. However, the outcome of
species interactions ultimately depends on population-level processes. Therefore, patterns of
species segregation and aggregation might be more clearly expressed in abundance matrices, in
which the matrix entries indicate the abundance or density of a species in a particular site.
We conducted a series of benchmark tests to evaluate the performance of 14 candidate null
model algorithms and six covariation metrics that can be used with abundance matrices. We
first created a series of random test matrices by sampling a metacommunity from a lognormal
species abundance distribution. We also created a series of structured matrices by altering the
random matrices to incorporate patterns of pairwise species segregation and aggregation. We
next screened each algorithm–index combination with the random and structured matrices to
determine which tests had low Type I error rates and good power for detecting segregated and
aggregated species distributions. In our benchmark tests, the best-performing null model does
not constrain species richness, but assigns individuals to matrix cells proportional to the
observed row and column marginal distributions until, for each row and column, total
abundances are reached.
Using this null model algorithm with a set of four covariance metrics, we tested for patterns
of species segregation and aggregation in a collection of 149 empirical abundance matrices and
36 interaction matrices collated from published papers and posted data sets. More than 80% of
the matrices were significantly segregated, which reinforces a previous meta-analysis of
presence/absence matrices. However, using two of the metrics we detected a significant pattern
of aggregation for plants and for the interaction matrices (which include plant–pollinator data
sets). These results suggest that abundance matrices, analyzed with an appropriate null model,
may be a powerful tool for quantifying patterns of species segregation and aggregation.
Key words: abundance matrix; biogeography; co-occurrence; covariation; null model; passive sampling;
statistical test.
INTRODUCTION
A major research focus in ecology has been the
elucidation of community assembly rules, a set of
mechanisms that lead to nonrandom patterns in
multispecies assemblages (Weiher and Keddy 1999).
For example, Diamond (1975) hypothesized that pairs
of species that are close competitors may never coexist in
the same local assemblage, leading to a biogeographic
‘‘checkerboard distribution.’’ Patterson and Atmar
(1986) hypothesized that orderly extinction sequences
in fragmented habitats will lead to a pattern of species
‘‘nestedness’’ (Ulrich et al. 2009). More recent assembly
rules have been based on patterns of phylogenetic
clustering or overdispersion (Webb et al. 2002, Emerson
and Gillepsie 2008), patterns of species interaction
Manuscript received 20 November 2009; revised 25 March
2010; accepted 30 March 2010. Corresponding Editor: M.
Fortin.
3 E-mail: [email protected]
networks (Bascompte and Jordano 2007), and the
distribution of species-level morphological or physiological traits (Bellwood et al. 2002, Lavorel and Garnier
2002).
Assembly rules are rarely tested experimentally (e.g.,
Fukami and Morin 2003, Irving and Connell 2006) and
are controversial because different mechanisms, including stochastic processes, may lead to the same community pattern (Gotelli 2004). Null models provide a
statistical test for whether an observed pattern is likely
in the absence of a particular mechanism (Gotelli and
Graves 1996), and they have always figured prominently
in the assembly rules literature (Williams 1964, Harvey
et al. 1983). The data for null model analyses are
typically in the form of a binary presence/absence matrix
(McCoy and Heck 1987): rows are species, columns are
sites or samples, and the entries indicate the absence (0)
or presence (1) of a species. In interaction or food web
matrices, both rows and columns may represent species,
and the entries represent the absence (0) or presence (1)
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of an interaction link, such as a trophic or pollinator
interaction (Jordano et al. 2003).
Although assembly rule patterns can be quantified
from presence/absence data, colonization or extinction
is usually preceded by changes in population size.
Colonization and extinction thus represent special cases
of change in abundance. An abundance matrix contains
rows as species, columns as sites or samples, and entries
representing the population size (which may be 0) of a
particular species in a particular site. Presence/absence
matrices can always be constructed from abundance
matrices, but not vice versa. The pattern of abundances
across replicated assemblages may potentially contain a
more complex and subtle signal of community assembly
rules than binary presence/absence matrices. Because
many assembly rules are based on the premise of species
interactions (Weiher and Keddy 1999), an appropriate
null hypothesis for abundance data might be that the
covariances between pairs of species equal zero (Schluter
1984). However, testing this null hypothesis with simple
and partial correlation analysis is problematic because
of constraints on the magnitude of correlations and
covariances (Brown et al. 2004).
Null models and randomization methods may be
preferable for the analysis of abundance matrices
because they are not as restrictive in their assumptions
as standard parametric statistics (Manly 1991). However, until recently, relatively little attention has been paid
to assembly rule patterns in abundance matrices (Graves
and Gotelli 1993, Hausdorf and Hennig 2007, Lester et
al. 2009).
In this paper, we explore null model analyses of
abundance matrices. We present six candidate indices
that may be used to describe the structure of abundance
matrices, and we test each index with 14 potential null
model algorithms that randomize the pattern in an
observed abundance matrix. These benchmark tests
were applied to a series of artificial matrices that were
simulated from distributions with specified patterns of
randomness and structure. Finally, following the example of several meta-analyses of binary presence/absence
matrices (Gotelli and McCabe 2002, Blüthgen et al.
2007, Ulrich and Gotelli 2007a), we assembled from the
literature and the Internet 147 empirical abundance
matrices and 36 mutualistic interaction matrices and
analyzed them with a subset of null models that
performed best in our initial benchmark tests.
MATERIALS
AND
METHODS
Strategies for evaluation of null model algorithms
Before a new randomization test is applied to
empirical data, its performance needs to be evaluated
with artificial data sets that have specified amounts of
randomness and structure (Gotelli 2001). Two properties are desirable in a statistical test. First, when the test
is confronted with ‘‘random’’ matrices, it should not
reject the null hypothesis too frequently, and a
traditional Type I error criterion of 5% is usually
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employed. Second, when the test is confronted with
‘‘structured’’ matrices, it should not accept the null
hypothesis too frequently. The statistical power of the
test is the probability of correctly rejecting the null
hypothesis when it is false. There is no convention for
power levels, but a value of 0.8 (the null hypothesis is
correctly rejected 80% of the time) has been suggested
(Cohen 1992). However, power analyses are rarely
conducted in ecological studies (Toft and Shea 1983),
perhaps because they require specification of an
alternative hypothesis and an effect size that can be
detected by the test.
In the context of null model analysis, what constitutes
a ‘‘random’’ or a ‘‘structured’’ matrix? The primary goal
in much null model analysis has been testing for the
effects of species interactions on community patterns
(Gotelli and Graves 1996). So, a ‘‘random’’ matrix
would consist of repeated samples from an assemblage
that is not structured by species interactions. Unfortunately, we can only speculate what such assemblages
would look like (Colwell and Winkler 1984).
Three approaches have been used to create ‘‘random’’
binary matrices for the purposes of benchmarking the
performance of null models. First, Gotelli (2000) began
with an empirical presence/absence matrix and created a
mixture of random matrices by uniformly reshuffling
elements within the matrix, sometimes constraining the
reshufflings within each row or column of the original
matrix. The disadvantages of this method are that it is
somewhat arbitrary, that it may include some of the
same null model algorithms that are being tested, and
that the results may be conditional on a particular
matrix size, dimension, or percentage fill. A second
approach is to specify a mechanistic colonization model
that does not include species interactions, such as the
neutral model (Bell 2005), and then use that model to
create random matrices that can be used to evaluate null
model procedures (Ulrich 2004). The disadvantage of
this method is that the test is narrowly optimized for one
particular mechanistic model, and there is no logical
reason that this model should have priority. Moreover,
even the simplest mechanistic model may contain several
parameters and specifying or estimating those parameters from real data is not easy (Gotelli and McGill 2006).
A third approach, which we have used here and in our
other studies (Ulrich and Gotelli 2007a, b, Gotelli and
Ulrich 2010), is to create unstructured matrices by
randomly sampling from a specified statistical distribution. This distribution should have two properties: first,
matrices that are created in this way should be similar to
empirical matrices in ‘‘external’’ properties such as their
marginal totals of species richness (number of species
occurrences per site) and species occurrence (number of
site occurrences per species), but should be random with
respect to the ‘‘internal’’ structure of the matrix (cooccurrence and covariance in abundance). Second,
sampling from this statistical distribution should mimic
a generic process model of random, independent
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colonization. However, like a nonparametric test, our
random sampling algorithm does not specify the
colonization and extinction details of a true process
model. With this strategy, the hope is that the results will
be more general and capture stochastic patterns that
might arise from a variety of different process-based
models.
To meet these criteria (realistic marginal distributions
and mimicry of a process model), we have created
random matrices by drawing samples from a lognormal
species abundance distribution. This distribution captures a property that has been observed in many real
assemblages: a small number of species are very
common, but most species are very rare (Preston
1962). When species are ranked according to their
abundance or occurrence, this generates a characteristic
right-skewed histogram that is approximated by sampling from a lognormal distribution. Whether the
lognormal distribution itself is caused by species
interactions or reflects neutral processes is still open to
debate (May 1975, Sugihara 1980, McGill et al. 2006),
but abundance and occurrence data collected for many
taxa at widely different spatial scales often conform to
an approximate lognormal distribution (McGill et al.
2007, Ulrich et al. 2010).
The distribution of species richness (species per site) is
more problematic because it depends largely on the
spatial grain and extent of sampling, which are often
determined by the investigator. For island archipelagos,
there may be a large amount of heterogeneity in the
abundance and number of species per site, much of
which is correlated with island area (Williamson 1981).
In contrast, for small-scale samples of fixed area in
homogeneous habitat, there may be relatively little
variance in species richness per site. Without any a
priori guide to modeling the species richness distribution, we used a random uniform distribution to
determine the total abundance (number of individuals)
per site and then sampled the individuals (and species)
from the lognormal distribution. Although the formal
definition of a Type I error is incorrect rejection of a true
null hypothesis, we use an operational definition here of
rejection of H0 on a set of appropriate test matrices
created by random sampling from a lognormal distribution.
Just as it is challenging to specify community patterns
in the absence of species interactions, it is equally
challenging to specify the patterns that would be
expected if communities were organized by strong
species interactions. Formal mathematical theory is of
little help here, because even simple models with
appropriate parameter values can generate virtually
any quantitative pattern of abundance and co-occurrence (Pielou 1981). However, the assembly rules
literature (and much of the statistical analysis of
empirical community structure) has emphasized that
negative species interactions will lead to segregated
patterns of species occurrence, including missing species
Ecology, Vol. 91, No. 11
combinations and checkerboard distributions (Diamond
1975), species exclusion by strong predator effects
(Morin 1983), and phylogenetic overdispersion of
species that do co-occur (Emerson and Gillespie 2008).
Aggregated patterns of co-occurrence can arise through
species mutualisms and positive interactions, guild
structuring, ordered extinction or differential colonization, and habitat filtering (see review in Ulrich et al.
[2009]).
Therefore, a worthwhile strategy for evaluating null
model algorithms is to measure their ability to detect
these simple patterns. For example, Gotelli et al. (1997)
began with a highly structured artificial matrix (first
proposed by Diamond and Gilpin [1982]) that contained
numerous segregated species pairs that never occurred
together in the same site (checkerboard distributions).
These kinds of extreme patterns are easily detected by
most null model tests. Next, Gotelli et al. (1997)
randomly swapped a few elements within some rows of
the matrix, which has the effect of partially randomizing
some of the species occurrences. The null model tests are
applied to the new matrix, and the procedure is
repeated, progressively adding more noise to the matrix.
This procedure is analogous to taking a new (ordered)
deck of playing cards, swapping two of the cards, and
then asking a naive observer to inspect the deck after
each swap and decide whether it has been shuffled or
not. As more and more noise is progressively added to
the deck through additional swaps, at some point it
becomes impossible to recognize the ‘‘structure’’ that
was present in the original ordering. Gotelli (2000) used
this method and found that the fixed–fixed algorithm for
presence/absence analysis could still detect significant
patterns when ;50% of the original checkerboard
matrix had been reshuffled.
In this study (and in Ulrich and Gotelli [2007a] and
Gotelli and Ulrich [2010]), we have taken the opposite
approach: we begin with a ‘‘random’’ matrix and then
increase the numbers of segregated or aggregated species
pairs within it and ask whether the null model algorithm
can detect the change. This procedure directly addresses
one of the early criticisms of null model analysis:
matrices with a small number of strongly interacting
species pairs might appear random because the interactions cannot be detected in a large matrix with many
noninteracting species pairs (the ‘‘dilution effect’’;
Diamond and Gilpin 1982). Additional analysis of
individual species pairs can also help to pinpoint which
particular pairs are contributing to nonrandomness of
the pattern for the entire matrix (Gotelli and Ulrich
2010).
It may be impossible to generate an optimal test
because of the inevitable trade-off between Type I and
Type II statistical errors. In our analyses, we have
placed a greater priority on minimizing Type I errors
(incorrect rejection of a true null hypothesis; ShraderFrechette and McCoy 1992). There are two reasons for
this. First, the entire null models controversy originat-
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COVARIATION ANALYSIS OF ABUNDANCE DATA
ed precisely over the question of Type I errors: did
apparently unusual biogeographic patterns in species
occurrence imply the existence of strong species
interactions and assembly rules or might they have
arisen by chance (Connor and Simberloff 1979)?
Second, most of the data sets that have been used in
null model analysis are ‘‘natural experiments’’ and are
not based on controlled field manipulations. In such
cases, the inference of mechanism from pattern is
always weaker, so we prefer a more conservative
approach that minimizes Type I errors. Therefore, we
first confronted our candidate algorithms and metrics
with the ‘‘random’’ matrices to eliminate tests with high
Type I error rates. Then we evaluated a subset of
algorithms and metrics for their performance on a set
of ‘‘structured’’ matrices.
In spite of our attempt to test a broad array of null
models and algorithms, these analyses are still optimized
for their performance on the set of matrices that we
created by random sampling from a lognormal distribution of species abundances. The tests are not fail-safe;
it is certainly possible to generate matrices with a model
of species interactions that would be incorrectly
classified as random (Colwell and Winkler 1984) or,
conversely, to generate matrices with a model of a
stochastic process (Ulrich 2004) that would be incorrectly classified as nonrandom. But our benchmark
analyses at least provide insight into how these analyses
will perform with a set of artificial matrices that
resemble real data in many respects and whose
properties are known. In the future, perhaps it will be
possible to tailor a particular test algorithm to a
particular empirical matrix for maximum power. Recently, Ladau (2008) has proposed optimal null model
tests based on formal parametric statistical theory.
These alternative procedures are promising, although
they are vulnerable to most of these same criticisms
discussed here and are not as transparent as traditional
null model analysis.
Matrix structures
We simulated two types of random abundance
matrices (200 matrices each) to study the properties of
14 randomization algorithms and six measures of
covariation (Fig. 1). Additionally we used 185 empirical
matrices compiled from the literature that contained
abundance data to apply the best performing randomization algorithms and measures and to infer the
frequency of nonrandom species associations. Of these
matrices, 149 were standard abundance matrices (rows ¼
species, columns ¼ sites) and 36 of these matrices were
interaction matrices (rows, columns ¼ species).
Random matrices.—We created 200 matrices (MR) by
assigning individuals randomly to matrix cells. The
number of columns (¼ sites) in each matrix was
determined by sampling from a random uniform
distribution (5 n 50 sites). To determine the number
of rows (¼ species), we first set the total number of
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species in the metacommunity ST by sampling from a
random uniform distribution (10 ST 200 species).
However, not all of these species will necessarily be
represented in the matrix because some rare species will
be missing due to insufficient sampling. To mimic these
sampling effects (the ‘‘veil line’’ of the lognormal
distribution; Preston 1962), we first specified the total
abundance Ni of species i by sampling from a lognormal
distribution of abundances:
Ni ¼ exi =2a
ð1Þ
where xi ; N(0, 1) and a is a shape-generating parameter
for each matrix that is sampled from a continuous
uniform distribution (0.1 a 1.0). A recent metaanalysis (Ulrich et al. 2010) confirmed that the
lognormal distribution most often provides the best fit
to abundance data sampled from natural communities.
The distribution of Ni is actually a scale mixture of a
lognormal and a uniform distribution, with a ¼ 0.5
generating a standard lognormal. For large, wellsampled communities, the value of a is often ;0.2
(Preston 1962, May 1975). To mimic the frequently
observed lower truncation of the lognormal (the veil
line), we sorted the ST species according to decreasing
abundance and used only the Smax most abundant
species in which the cutoff point ‘‘max’’ was sampled
from a random uniform distribution (ST/2 Smax ST ). This sampling procedure resulted in a matrix with
Smax rows and n columns.
To mimic different carrying capacities per site, the
relative abundances Aj for each site j were also drawn
from a random uniform distribution (0 , Aj 1.0).
Having established the relative abundances for the
columns and the rows of the matrix, we then assigned
individuals randomly to each cell in the matrix, with
the probability of choosing a particular row and
column being set proportional to the row and column
abundance totals (see Gotelli 2000). We placed
individuals randomly in this way until all Smax species
were represented by at least one individual in at least
one site. The relative abundance distributions generated in these matrices reflected a variety of patterns that
can arise by sampling from a lognormal distribution
and are similar to abundance distributions measured in
nature (Magurran 2004, McGill et al. 2007, Ulrich et al.
2009).
Because the method of placing individuals in the
matrix might potentially affect the assemblage patterns,
we created 200 additional random matrices (MS), using
a slightly different algorithm. As before, the matrix
dimensions were established by random uniform draws
to determine the number of columns (5 n 50) and
the maximum number of species in the metacommunity
(10 ST 200), again sampling from a truncated
lognormal abundance distribution. However, for the MS
matrices, we placed individuals sequentially (site by site)
until for each site j the designated number of species Sj
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Ecology, Vol. 91, No. 11
FIG. 1. Flowchart illustrating benchmark testing of covariation metrics (CA to Mantel) and null model algorithms (IA to PC)
against two sets of random matrices (random, MR, and sequential, MS) and two sets of seeded matrices (Mmod), and testing of
empirical matrices. See Materials and methods: Strategies for evaluation of null model algorithms and Covariance metrics for
explanations of abbreviations.
was achieved. Sj was determined by random sampling
from a uniform distribution (1 , Sj ST ).
Empirical matrices.—We also analyzed 149 species 3
sites abundance matrices collected from the literature;
all matrices used in our analyses are found in the
Supplement. We classified matrices according to the
taxon studied (mammals, birds, fish, arthropods, nonarthropod invertebrates, and plants), biome (terrestrial,
aquatic), and, for terrestrial studies, habitat (mainland,
island). We used only matrices that were based on
quantitative sampling and provided integer counts of
abundance (no biomass measures or ranked abundances). We analyzed separately 36 interaction matrices
(species 3 species) that included abundance data from
the National Center for Ecological Analysis and
Synthesis (NCEAS) database (available online4; see also
the Supplement). In 53 of the data sets, the entries were
density rather than abundance. We created abundance
matrices in these cases by assuming the rarest species in
the matrix was represented by a single individual.
4 hhttp://www.nceas.ucsb.edu/interactionweb/html/
datasets.html#anemone_fishi
Although multiple matrices were used from some
studies, these almost always were for samples of
different taxa or different sites. Twenty of the matrices
were of the same taxa (studies on seasonal species
turnover of spiders, ground, and rove beetles) sampled
in different times. Therefore, we treated each empirical
matrix as an independent observation in our analyses of
summary patterns.
Covariance metrics
We developed and analyzed six indices that quantify
the pattern of species aggregation and segregation in
abundance matrices; some of these are directly analogous to co-occurrence metrics that are commonly used
in the analysis of presence/absence matrices.
1) We used an abundance analog of ‘‘checkerboard’’
distributions (Diamond 1975). In presence/absence
matrices, ‘‘checkerboard units’’ (Stone and Roberts
1990) represent submatrices of the following form:
1
0
0
1
(rows and columns of this submatrix form do not have
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COVARIATION ANALYSIS OF ABUNDANCE DATA
to be adjacent). The more checkerboard units there are
in a matrix, the more segregated species are in their
occurrence. We define an ‘‘abundance checkerboard’’ as
a 2 3 2 submatrix of the form
a b
a.b a.c d.b d.c
c d
or
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5) Chao et al. (2008) extended the Morisita index of
similarity for two communities to a matrix-wide metric
for n communities of the following form:
"
!2
#
m
n
n
X
X
X
2
pij ð pij Þ
MA ¼
i¼1
j¼1
j¼1
n
X
ðn 1Þ ðpij Þ2
ð7Þ
j¼1
a,b a,c d,b
d,c
ð2Þ
where a, b, c, and d represent the abundances of two
species in two different sites. The metric CA is a count of
the total number of abundance checkerboards in the
matrix. This metric can be standardized with regard to
matrix size (m rows, n columns) by
CAST ¼
4CA
:
mðm 1Þnðn 1Þ
ð3Þ
The standardized CA value can range from 0.0 to 1.0,
with high values of CA indicating more negative
covariation in abundances.
2) Similarly, we define the number of species
abundance aggregations AA as a count of aggregated
2 3 2 submatrices of the form
a b
a.b a.c d,b d,c
c d
or
a,b
a,c
d . b d . c:
ð4Þ
Again a standardized metric has the following form:
AAST ¼
4AA
:
mðm 1Þnðn 1Þ
ð5Þ
The standardized AA value can range from 0.0 to 1.0,
with high values of AA indicating positive covariation in
abundance of species. AA and CA are correlated, but
may differ in their ability to detect positive or negative
covariation.
3) Rather than just counting abundance checkerboards as CA and AA, we can quantify the strength of
covariance from the differences in abundance of the
checkerboard elements:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X ða cÞ2 þ ðb dÞ2
4
a2 þ b2 þ c2 þ d2
:
ð6Þ
AST ¼
mðm 1Þnðn 1Þ
Large values of AST suggest strong positive covariance
among species.
4) The variance test of Schluter (1984) is a metric of
species covariance in abundance. This test compares the
variance of row totals V with the sum of the column
variances W. If the average covariance in abundance
among all pairs of species ;0.0, the value U ¼ V/W
should be v2 distributed with n degrees of freedom. Low
values of U indicate negative covariation in abundance.
where pij is the relative abundance of species i in site j.
Low values of MA indicate dissimilarity of species
relative abundance distribution among sites, which can
be interpreted as a measure of negative covariance in
relative abundances.
6) The Mantel test identifies nonrandom correlations
between two matrices (Mantel 1967). To assess whether
the MR matrices were nonrandom, we used the mean
Mantel correlation between the MR matrix (using the
Pearson correlation as distance metric) and the matrices
generated by different null model algorithms. The
expected correlation and confidence limits came from
100 randomly sampled null matrices. The Mantel test
used in this way can indentify nonrandomness, but it
does not indicate whether an observed matrix is
unusually aggregated or segregated.
To compare the performance of the above metrics to
those that are commonly used for presence/absence
analyses, we used the C score (CS; Stone and Roberts
1990) as a measure of species segregation and the
discrepancy metric BR (Brualdi and Sanderson 1999) as
a measure of species nestedness (Ulrich et al. 2009).
Null model algorithms
Compared to the analysis of binary presence/absence
matrices (Gotelli 2000), there are many more possible
algorithms and constraints that can be used to
randomize an abundance matrix. Null model algorithms
for abundance matrices can be divided into individualbased and population-based algorithms. Individualbased algorithms randomize the placement of individuals in the matrix. Population-based algorithms preserve
population abundance values and randomize their
occurrences among sites or species. Individual-based
algorithms can be further divided into fixed-zero and
floating-zero algorithms. Fixed-zero algorithms preserve
species occurrences, so that the pattern of presences and
absences (but not abundances) in the null matrices
match those in the original matrix. Floating-zero
algorithms allow for the placement of individuals in
matrix cells that contained zeroes in the original matrix.
We did not analyze any population-based floating-zero
algorithms.
We used three population-based fixed-zero algorithms:
1) PM reshuffles populations equi-probably among
the nonempty cells of the entire matrix. This model
alters row and column abundance totals but preserves
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WERNER ULRICH AND NICHOLAS J. GOTELLI
species occurrences and the grand total of abundances
for the entire matrix. Following the recommendation of
Lehsten and Harmand (2006), we used 100 3 n 3 m such
reshufflings to generate each null matrix for the PM, PC,
and PR algorithms.
2) PC reshuffles populations equi-probably among the
nonempty cells of each column. This model preserves
species occurrences and total abundance per site, but
alters abundances of each species.
3) PR reshuffles populations equi-probably among the
nonempty cells of each row. This model preserves
species occurrences and total abundance per species,
but alters abundances of each site.
We used two individual-based fixed-zero algorithms:
1) OS first clears the matrix to be tested and then
assigns individuals randomly only to cells that originally
had nonzero values. The probability of placing an
individual in a particular cell is proportional to the
observed row and column abundance totals for that cell.
Individuals are assigned sequentially to the matrix in
this way until the total number of individuals in the
original matrix is reached. OS preserves species occurrences, but does not preserve observed row and column
total abundances.
2) OF first clears the matrix to be tested and then
assigns individuals randomly only to cells that originally
had nonzero values. The probability of placing an
individual in a particular cell is proportional to the
observed row and column abundance totals for that cell.
Individuals are assigned sequentially to the matrix in
this way until, for each row and column, total
abundances are reached. This algorithm allows the
abundance in each cell to vary, but preserves both
species occurrences and row and column abundance
totals of the original matrix. In a few cases, this
algorithm stopped placing individuals before the total
abundances were reached because the simultaneous
constraints on row and column totals could not be
met. However, the total number of individuals that
could not be placed was always less than 10 (,0.1%) and
should not affect the performance of the test.
We used nine individual-based floating-zero algorithms:
1) IR assigns individuals randomly to matrix cells
with probabilities proportional to observed row and
column abundance totals until total species richness is
reached. In a few cases, this algorithm generated
matrices with empty columns (sites), which were
discarded prior to analysis.
2) IS assigns individuals randomly to matrix cells with
probabilities proportional to observed row and column
abundance totals until the total number of occurrences
is reached for each row and column.
3) ISR sequentially (row by row) assigns individuals
randomly to each row with probabilities proportional to
observed column abundance totals until the respective
number of row occurrences is reached.
Ecology, Vol. 91, No. 11
4) ISC sequentially (column by column) assigns
individuals randomly to each column with probabilities
proportional to observed row abundance totals until the
respective column total species richness is reached.
5) IT assigns individuals randomly to matrix cells with
probabilities proportional to observed row and column
abundance totals until, for each row and column, total
abundances are reached.
6) ITR sequentially (row after row) assigns individuals randomly to each row with probabilities proportional to observed column abundance totals until the
respective row total (the number of individuals) is
reached.
7) ITC sequentially (column after column) assigns
individuals randomly to each column with probabilities
proportional to observed row abundance totals until the
respective column total (the number of individuals) is
reached.
8) IA reassigns all individuals randomly to matrix
cells with probabilities proportional to observed row
and column abundance totals until the matrix-wide total
number of individuals is reached. In a few cases, this
algorithm generated matrices with empty rows (species)
or columns (sites), which were discarded prior to
analysis.
9) IF is a two-step algorithm that preserves row and
column abundances and species richness. In the first
step, the algorithm converts the abundance matrix into a
presence/absence matrix. Using a standard swap procedure (Gotelli 2000), 2 3 2 submatrices of the form
0 1
1 0
or
1 0
0 1
to
0 1
1 0
or
1 0
0 1
are reshuffled, again using 100 3 n 3 m reshufflings. In
the second step, the nonzero cells are cleared and then
filled according to the OF algorithm.
All null models and indices were calculated with the
software applications CoOccurrence and Matrix (see
Supplement).
Diagnostic tests
We first determined, for each combination of null
model (14 algorithms) and covariation index (six
indices), whether the null model correctly identified
most of the random matrices as being random. For each
algorithm–index combination, we estimated the tail
probabilities for the set of 200 random MR test matrices
by simulating 1000 null assemblages for each random
matrix. For the MS test matrices, we also simulated 1000
null assemblages, but used only the four most promising
algorithm–index combinations (Fig. 1), based on their
performance with the MR test matrices. If the analysis is
not prone to Type I statistical errors, then approximate-
November 2010
COVARIATION ANALYSIS OF ABUNDANCE DATA
3391
TABLE 1. Numbers of random matrices (MR; from a total of 200) identified by six metrics of species covariation as being either
segregated of aggregated (using the upper [UCL] and lower [LCL] 97.5% confidence limits).
CA
SA
AA
MA
Mantel
U
Algorithm
LCL
UCL
LCL
UCL
LCL
UCL
LCL
UCL
LCL
UCL
LCL
UCL
IA
IT
ITC
ITR
IS
ISC
ISR
IR
IF
OA
OF
PM
PR
PC
0
20
1
3
43
1
90
65
8
58
0
182
171
180
39
1
29
14
25
34
68
6
11
2
72
3
1
1
0
17
1
2
38
1
7
50
10
40
0
183
173
183
40
2
26
12
19
30
174
15
12
3
88
2
0
1
6
2
3
9
1
42
189
113
54
77
12
0
0
0
10
34
14
5
89
0
9
0
0
1
4
191
174
191
5
0
10
2
0
10
131
91
25
5
8
0
1
0
8
21
8
8
39
15
66
0
12
48
24
194
145
193
3
2
8
4
0
1
94
73
3
2
20
0
1
0
2
20
4
6
98
80
74
4
45
17
56
196
181
193
0
0
0
0
0
4
189
49
0
0
0
0
15
0
0
0
0
0
2
0
0
0
0
166
164
158
142
158
Note: See Materials and methods: Strategies for evaluation of null model algorithms and Covariance metrics for explanations of
abbreviations.
ly five of the 200 test matrices should be statistically
significant in the upper tail (P . 0.975) and five should
be significant in the lower tail (P , 0.025) of the
distribution.
After discarding a large number of model–index
combinations that failed this test, we then studied the
statistical power of the most promising combinations. We
did this by modifying the 200 MR matrices and generating
from each of them four new matrices (Mmod). First, we
generated 600 matrices for which 0.01% to maximally
33% randomly selected aggregated abundance checkerboards (as defined by Eq. 4) were rearranged as
segregated checkerboards (as defined by Eq. 2). We then
generated 200 matrices for which 0.01–33% randomly
selected segregated abundance checkerboards (Eq. 2) of
each MR matrix were changed into aggregated checkerboards. Because such changes potentially alter many
other checkerboards in the matrix, the resulting number
of segregated or aggregated checkerboards ranged from 0
to 7% of the total number of 2 3 2 submatrices [nm(n 1)(m 1)/4]. We then calculated the fraction of these
modified matrices that were correctly identified as
statistically significantly segregated or aggregated (P ,
0.05 in either tail) by the different algorithms. From these
analyses, we were able to identify combinations of
algorithms and metrics that had the best power for
detecting segregated and aggregated distributions from
nonrandom matrices. Naturally, greater replication in
benchmark tests would be desirable, but the tests are
time-consuming, and the results were clear-cut with these
samples of 200 and 600 test matrices.
We also calculated a standardized effect size (SES) as
a Z-transformed score [Z ¼ (x l)/r], where x is the
observed index for the MR or MS test matrix, l is the
mean of the 1000 simulated indices for each of the null
model algorithms, and r is the standard deviation of the
1000 simulated indices. The use of SES is based on the
assumption of an approximately normal error distribution. This was indeed the case: the mean skewness of all
null model distributions was only 0.004 with a standard
deviation of 0.37. For a random sample of scores that
follows a normal distribution, ;95% of the SES values
should be ,j2.0j. We used the SES to test whether null
model results were sensitive to basic matrix properties
(size, fill, and mean species abundances).
Based on the results of these tests, we used one of the
algorithms (IT) in combination with four metrics (MA,
U, CA, SA) to test for patterns of segregation and
aggregation in the empirical matrices.
RESULTS
Performance of null model algorithms
with random test matrices
Null models that reshuffled whole populations (PM,
PR, PC) performed poorly (Table 1). Irrespective of
metric, they identified more than 70% of the random MR
matrices as being segregated. The two null models that
resampled observed occurrences (OA, OF) identified
between 10% (OA with the variance test) and 83% (OA
with the Mantel test) of the test matrices as nonrandom.
Individual-based null models that were conditioned on
row, column, or total abundances (IA, IT, ITC, ITR)
performed better than those that were constrained to
match observed species richness (IS, ISC, ISR, IR). In
particular, the individual-based null models IT, ITC,
and ITR identified only 0–15% of the MR matrices as
being not random (Table 1). The performance of the
two-step IF null model was a bit worse: 0–27% of
random MR matrices were statistically significant. For
the sequentially filled random MS matrices, IA, ITC,
and ITR performed worse and incorrectly classified 0–
79% as not random (Table 2). Overall, the best
performing null model was IT, which identified a
maximum of 17% of the MR matrices (with the AA
metric) and 13% of the MS matrices (with the Mantel
test) as being not random.
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WERNER ULRICH AND NICHOLAS J. GOTELLI
Ecology, Vol. 91, No. 11
TABLE 2. Numbers of sequential matrices (MS; from a total of 200) identified by six metrics of species covariation as being either
segregated or aggregated (using the upper [UCL] and lower [LCL] 97.5% confidence limits), using only the most promising null
models of Table 1.
CA
SA
AA
MA
Mantel
U
Algorithm
LCL
UCL
LCL
UCL
LCL
UCL
LCL
UCL
LCL
UCL
LCL
UCL
IA
IT
ITC
ITR
0
3
0
0
133
9
69
84
0
1
0
0
133
11
72
83
94
5
8
82
0
2
2
0
20
0
61
2
2
11
3
67
64
0
69
1
25
13
3
37
0
26
0
0
0
1
0
158
Note: See Materials and methods: Strategies for evaluation of null model algorithms and Covariance metrics for explanations of
abbreviations.
Performance of covariance metrics
with random test matrices
For the best performing null models IA, IT, ITC, and
ITR, many of the six metrics tested had satisfactory Type
I error rates and correctly classified .77% of the MR
matrices as being random (Table 1). For the MS matrices,
all metrics except the Mantel test performed well with the
IT null model, but largely failed with IA, ITR, and ITC
(Table 2). Results of null model tests using the metrics
MA, AA, and U were least dependent on matrix size, fill,
and mean abundance (Table 3). The Z-transformed
values of CA, SA, and, particularly, the Mantel test
TABLE 4. Proportions of modified matrices (Mmod; total of
800) that were detected as being either aggregated (1% to
10% checkerboards, given as a percentage of the total
number of 2 3 2 submatrices) or segregated (0.01% to 10%).
Change in number
of checkerboards
IA
IT
ITC
ITR
CA
1% to 10%
0.01% to 0.1%
0.1% to 1%
1% to 10%
0.05
0.25
0.53
0.94
0.45
0.02
0.11
0.81
0.06
0.17
0.43
0.94
0.27
0.10
0.35
0.89
TABLE 3. Pearson correlation coefficients between matrix size,
fill, and mean abundance (¼ mean number of individuals per
cell), and the Z-standardized covariation metric for the four
best-performing null models (Table 1) of the random
matrices (MR).
SA
1% to 10%
0.01% to 0.1%
0.1% to 1%
1% to 10%
0.06
0.27
0.50
0.93
0.46
0.01
0.10
0.76
0.08
0.16
0.40
0.94
0.28
0.11
0.30
0.87
Matrix and properties
AA
1% to 10%
0.01% to 0.1%
0.1% to 1%
1% to 10%
0.09
0.07
0.25
0.80
0.13
0.03
0.13
0.72
0.06
0.16
0.43
0.91
0.06
0.09
0.33
0.84
CS
1% to 10%
0.01% to 0.1%
0.1% to 1%
1% to 10%
0.08
0.26
0.39
0.91
0.44
0.01
0.06
0.69
0.10
0.16
0.30
0.90
0.30
0.07
0.20
0.79
MA
1% to 10%
0.01% to 0.1%
0.1% to 1%
1% to 10%
0.03
0.09
0.36
0.90
0.15
0.12
0.36
0.91
0.02
0.12
0.46
0.94
0.06
0.09
0.33
0.84
1% to 10%
0.01% to 0.1%
0.1% to 1%
1% to 10%
0.02
0.03
0.25
0.87
0.14
0.06
0.27
0.89
0.02
0.08
0.35
0.90
0.13
0.05
0.28
0.87
Mantel
1% to 10%
0.01% to 0.1%
0.1% to 1%
1% to 10%
0.00
0.02
0.14
0.54
0.00
0.01
0.05
0.42
0.00
0.02
0.12
0.51
0.00
0.01
0.09
0.41
BR
1% to 10%
0.01% to 0.1%
0.1% to 1%
1% to 10%
0.05
0.20
0.19
0.76
0.29
0.02
0.05
0.58
0.05
0.14
0.13
0.74
0.25
0.07
0.11
0.70
IA
IT
ITC
ITR
CA
Size
Fill
Mean abundance
0.00
0.44
0.31
0.16
0.26
0.14
0.02
0.30
0.23
0.09
0.31
0.15
SA
Size
Fill
Mean abundance
0.02
0.43
0.29
0.16
0.26
0.12
0.01
0.29
0.20
0.09
0.31
0.13
AA
Size
Fill
Mean abundance
0.00
0.23
0.10
0.17
0.11
0.09
0.00
0.10
0.02
0.03
0.11
0.07
MA
Size
Fill
Mean abundance
0.03
0.14
0.09
0.01
0.01
0.01
0.02
0.18
0.15
0.09
0.04
0.06
Size
Fill
Mean abundance
0.10
0.18
0.12
0.17
0.00
0.05
0.11
0.18
0.19
0.09
0.07
0.01
Mantel
Size
Fill
Mean abundance
0.04
0.54
0.27
0.05
0.27
0.21
0.06
0.36
0.13
0.08
0.41
0.21
U
Note: Significant correlations (P , 0.05) appear in boldface.
See Materials and methods: Strategies for evaluation of null
model algorithms and Covariance metrics for explanations of
abbreviations.
U
November 2010
COVARIATION ANALYSIS OF ABUNDANCE DATA
3393
FIG. 2. Fraction of 185 empirical abundance matrices that were significantly segregated (gray bars) or aggregated (white bars)
according to the metrics MA, U, CA, and SA under the IT null model: vertebrates (N ¼ 14), plants (N ¼ 3), non-arthropod
invertebrates (N ¼ 6), Carabidae (N ¼ 39), all arthropods (N ¼ 126), and interaction matrices (N ¼ 36). See Materials and methods:
Strategies for evaluation of null model algorithms and Covariance metrics for explanations of abbreviations.
tended to correlate with matrix fill. The largest correlation was between matrix fill and the Mantel value for the
IA algorithm (Pearson’s r ¼ 0.54, P , 0.01).
Performance of null model algorithms and covariance
metrics with segregated and aggregated test matrices
The diagnostic tests with matrices in which we
increased or decreased the numbers of abundance
checkerboards reveal the power of the null models to
correctly identify nonrandom patterns. These tests
suggest that the IT algorithm was less powerful than
IA, ITC, and ITR for detecting species segregation, but
was more powerful for detecting species aggregation
(Table 3). CA and SA in combination with IT had the
best power to detect species aggregation and correctly
identified 45% and 46%, respectively, of the manipulated matrices as aggregated (Table 4). The power of
both tests was comparable to the power of the C score
to detect aggregations when the same matrices were
analyzed in the form of presence/absence data. With
the IT algorithm, the abundance based metrics CA,
MA, and U correctly identified at least 81% of the
segregated matrices (1–10% increase in checkerboards)
as nonrandom. The presence/absence indices CS (C
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WERNER ULRICH AND NICHOLAS J. GOTELLI
Ecology, Vol. 91, No. 11
FIG. 3. Fraction of 149 empirical abundance matrices that were significantly segregated (gray bars) or aggregated (white bars)
according to the metrics MA, U, CA, and SA under the IT null model: aquatic (N ¼ 9), mainlands (N ¼ 139), and islands (N ¼ 23).
See Materials and methods: Strategies for evaluation of null model algorithms and Covariance metrics for explanations of
abbreviations.
score) and BR (nestedness) were more conservative and
identified only 69% and 58%, respectively, of these
matrices as segregated (Table 4). With the IT algorithm, the MA performed best (91% of segregated
matrices correctly identified), but the Mantel test was
too conservative (only 42% of the matrices identified).
Based on analyses of both random matrices (Tables 1–
3) and structured matrices (Table 4) the IT algorithm is
best for testing empirical matrices. The CA and SA
metrics are more conservative, and the MA and U
metrics are more liberal in the detection of nonrandom
patterns.
Meta-analysis of empirical abundance matrices
MA and U identified .80% of the 185 real abundance
matrices as being significantly segregated (Fig. 2). CA
and SA gave similar results, except that most plant and
interaction matrices were classified as random. However, CA and SA also identified aggregated patterns in the
plant and interaction matrices that were not detected
with MA and U. All four metrics identified .50% of the
aquatic, mainland, and island data set as being
significantly segregated (Fig. 3). Again, CA and SA
appeared to be more conservative than MA and U. The
C score, which is based only on presence/absence data,
was more conservative than the abundance-based
metrics and identified at most 39% of the animal and
interaction matrices as being segregated. However, this
score classified 33–81% of the matrices as being either
random or even aggregated (Fig. 4A) and the results did
not differ greatly among island/aquatic/mainland matrices (Fig. 4B).
DISCUSSION
Careful benchmark testing of potential randomization algorithms and community metrics is essential for
valid null model analyses (Gotelli 2001). In this case,
the vast majority of algorithms and metrics that we
evaluated had unacceptably high Type I error rates
when tested with a series of random matrices that were
created from random sampling of a lognormal species
abundance distribution (Table 1). In fact, the only
metric that met the strict criterion of rejecting H0 for
,5% of the null model matrices was the Mantel metric.
However, this index does not indicate whether a
nonrandom matrix is predominantly aggregated or
segregated, and it had poor power for detecting
significant aggregation or segregation (Table 4). Most
of the null models tests showed some correlations with
measures of matrix size or fill (Table 3). However, most
of the correlations are fairly weak, and they reflect the
universal property that, as sample size becomes very
large, the null hypothesis will inevitably be rejected
because the randomization algorithm is not identical to
the (lognormal) sampling model that was used to
generate the test matrices.
Overall, the IT algorithm performed best, with fairly
low Type I error rates (Tables 1 and 2) but good power
for detecting aggregated or segregated distributions with
a variety of metrics (Table 4). It is interesting to note
November 2010
COVARIATION ANALYSIS OF ABUNDANCE DATA
that the IT algorithm assigns individuals to matrix cells
proportional to observed row and column totals until,
for each row and column, total abundances are reached.
A similar null distribution underlies contingency table
analysis (Everitt 1980), in which the null hypothesis of
no species 3 site interaction is tested by assuming
independent marginal probabilities for each cell in the
matrix (cf. Diamond and Gilpin 1982). However, Monte
Carlo simulations usually do not produce identical
results to parametric tests of the same data (Gotelli
and Ellison 2004). In this case, a parametric chi-square
test (without corrections for small sample size or sparse
matrices; Gotelli and Ellison 2004) of the 200 random
MR matrices identified 33 (16.5%) as being not random
at the 5% error level, a value within the range observed
for the IT null model (Table 1). Alternative algorithms
are available that do not fix row and column abundance
totals, but allow them to vary among different simulated
matrices (see Gotelli and Graves [1996] and Gotelli
[2000] for a discussion of this algorithm for presence/
absence analysis). However, such tests are potentially
prone to greater Type I error, because the null
hypothesis might be rejected due to differences in row
and column sums per se, rather than because of
aggregated or segregated abundance distributions.
When we applied the IT test with several metrics to the
empirical abundance matrices, nearly all of them showed
strongly segregated patterns (Fig. 2), although a few
aggregated distributions were also detected for plant and
interaction matrices. Although differences among taxa
were strong (Fig. 2), differences among habitat type were
not (Fig. 2), which is similar to the findings of Gotelli
and McCabe (2002) for presence/absence matrices. The
frequency of segregated distributions in these real data
sets (Fig. 2) is far greater than would be expected from
the frequencies expected in our null model tests (Tables 1
and 2). When these same matrices were converted to a
presence/absence form and analyzed with the standard
fixed–fixed null model (Gotelli 2001), segregated patterns
still dominated, although the frequency of nonrandom
matrices was much lower (Fig. 4). These results suggest
that null model analysis of abundance matrices may
potentially be more powerful than null model analysis of
presence/absence matrices (Hausdorf and Hennig 2007),
although the latter are more common in the literature
and are easier for field biologists to generate. Detection
errors and imprecise counts are certainly present in both
abundance and presence/absence matrices, and they
potentially affect the power of the tests. However, these
factors have only recently been incorporated into
statistical tests for species interactions (Royle and
Dorazio 2008, Waddle et al. 2010).
Although ecologists routinely test for pairwise correlations of species abundances (Brown et al. 2004), there
has been relatively little use of null models with
abundance data. Some null model tests have been
applied to the analysis of relative abundance distributions (McGill et al. 2007), although these tests often use
3395
FIG. 4. Fraction of empirical abundance matrices (converted to presence/absence) with a significant C score (5% error
level) under the fixed–fixed null model classified (A) according
to taxon and type and (B) according to biome. In panel (B) only
the 147 species 3 site matrices are included. Gray bars indicate
species segregation; white bars indicate aggregation.
just the row sums of the species abundance matrix. Null
model tests have been used with species abundance
matrices in which the columns represent sampling
periods rather than sites. In the 1980s, ecologists used
null model tests of these data to determine whether
species ranks remained concordant through time (Grossman 1982, Ebeling et al. 1990), which is an important
measure of community, persistence, and stability (Pimm
1984). More recently, Houlahan et al. (2007) calculated
Schluter’s (1984) variance ratio (the U metric in our
analyses) for a large number of published species
temporal matrices. Most of these indices indicated a
pattern of species aggregation: abundances tended to
covary positively through time, suggesting that compensatory dynamics were not important. Schluter (1984)
found a similar pattern when he analyzed species 3 site
abundance matrices. However, the tests by Houlahan et
al. (2007) and Schluter (1984) assume that the columns
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WERNER ULRICH AND NICHOLAS J. GOTELLI
of the matrix (sites or times) are equivalent and do not
affect the probability of occurrence of individuals of
different species (Gotelli 2001). In contrast, the IT
algorithm that we used preserves the column totals for
abundance, taking into account differences in suitability
or conditions among sites. With this null model, the U
metric reveals mostly segregated patterns in species
abundance matrices (Fig. 2).
In summary, null model analysis that was first
developed for binary presence/absence matrices can be
effectively extended to abundance matrices and may be a
more powerful test for segregated, aggregated, and
random patterns of abundance. The statistical performance of null models against mechanistic colonization
and extinction scenarios is largely unknown, but some
insight can be gained by comparing the performance of
null models to simple sampling distributions, as we have
done here for the lognormal species abundance distribution. Like nearly all previous null models, our
algorithms do not use information on the location of
the samples or take into account spatial autocorrelation
(Lichstein et al. 2002). Most existing tests (including
ours) also treat abundance and presence/absence data as
error-free and do not address the problem of undetected
species (Dorazio 2007). Species detection and spatial
autocorrelation are promising, but largely unexplored,
avenues for future null model analyses.
ACKNOWLEDGMENTS
Comments by Lewi Stone, Robert Dorazio, and three
referees greatly improved an earlier version of this paper.
We thank Adam Barcikowski, Izabela Hajdamowicz, Piotr
Jastrzebski, Bartłomej Pacuk, Marzena Stańska, January
Weiner, and Marcin Zalewski for the permission to include
unpublished data sets in the present analysis. N. J. Gotelli
acknowledges support from grants NSF DEB-0541936 and
DOE DE-FG02-08ER64510. W. Ulrich was in part supported
by grants from the Polish Science Committee (KBN 3 P04F 034
22, and KBN 2 P04F 039 29).
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SUPPLEMENT
Data matrices and source codes of the software used in the paper (Ecological Archives E091-240-S1).